Preheat your oven to 350°F (175°C).
In a large mixing bowl, combine the sliced apples, granulated sugar, lemon juice, ground cinnamon, nutmeg, and flour. Toss until the apples are well coated.
Pour the apple mixture into the baking dish, spreading it evenly.
In another mixing bowl, whisk together the flour, sugar, baking powder, and salt for the cobbler topping.
Add the melted butter, milk, and vanilla extract to the dry ingredients. Stir until just combined; the batter will be somewhat lumpy.
Spoon the cobbler topping over the apple filling, ensuring it's evenly distributed.
Bake in the preheated oven for 40-45 minutes, or until the topping is golden and a toothpick inserted into the cobbler comes out clean.
Allow the cobbler to cool for about 10 minutes before serving.
Enjoy warm, optionally with a scoop of vanilla ice cream!